What local transportations (including buses or trains) to the following Harbin Attractions

1. Bus no. 29, 80, or 88 can take you to Sun Island, where you can enjoy the snow expo. The bus fare is 1-2RMB per person. If you go there during the ice and snow festival, the entrance ticket costs CNY240. The operating hours are 08:00-17:00 and 08:30-19:00 (during the snow sculpture expo).
2.Ice Lantern World, U may refer to the Ice Lantern Garden Party, isn’t it? If so, you can take bus no. 8, 53, 74, 83, 113, 114, 201 or 206 and get off at Zhaolin Park. The entrance ticket costs 200RMB.
3. Bus no. 88, 125 or 126 can directly take you to the Ice and Snow World, at the cost of CNY1-2. If you enter during 09:00 – 12:00, the fee is 150RMB. If you enter after 12:00, the fee is 300-330RMB.

4. Polarland: take bus no.29, 54, 80, 88 or 346 and get off at Sun Island and then walk there. The entrance ticket fee is 120RMB and the operating hours are 09:00-17:00.
5. Heilongjiang Provincial Museum: It is free of charge. The operating hours are from 09:00 to 16:00.
6. The church: You can use bus no. 1, 2, 13, 15, 64, 66 or 113 and the bus fare is CNY1-2. It costs CNY20 and opens from 08:00 to 18:00.
7. Central Street: it is just a street open for 24h. You can enjoy yourself freely and it is a must-go for your travel. Bus no. 53, 113, 114 or 201 would help.

The info above is what I know in the last. It may change a little this year.

Where can tourists from USA purchase admission tickets for the Attractions?

If you wanna book ticket from USA, I'm afraid you have no way but to turn to an agent or your Chinese friend. Actually, it is fine to buy tickets for all of them on the spot.

Thanks for your response.
I know you mention the buses but where can I ride these buses to these tourists attractions since I will be a new tourist arriving from USA to Harbin Airport ?

You are suggested to stay in a hotel in Central Street in the city. Buses to these attractions almost pass the Central Street.